Could coach show favoritism? Maybe. I know of at least a few instances I can think of. However, most coaches my kids have trained with can/have remained unbiased. One pro is coach knows you better, they've seen you play and know what work you need. I think it all depends on the coach.

I say it creates a conflict of interest and it leads to that player getting more games, more playing time, and just more opportunities. Coaches have egos and reputations to uphold and they will want to see that player make progress just a little bit more than a regular player. That may not become an issue with parents on a given team, but what happens if one or more parents from that team asks for privates too? I think it is better for the coach to not put themselves in that position.

No. Creates conflict that a position and coach can be bought. Whether true or not. It creates an undercurrent of jealousy on team. One caveat is that every once in a while okay if coach wants to work on something that benefits the team. But should be done sparingly and nothing consistent